When a \(\operatorname{CAT}(0)\)  space is complete and locally compact then the closure of the convex hull of a compact set is always compact. This is a direct corollary of Hopf–Rinow Theorem which states that closed and bounded sets are compact whenever the underlying geodesic metric space is complete and locally compact. Indeed if a set is compact then it must be bounded and closed, thus contained in a closed geodesic ball of a certain radius. Because geodesic balls in \(\operatorname{CAT}(0)\)  spaces are convex then the closure of the convex hull of the given compact set must certainly be included in this ball and consequently be compact.
While the problem seems to be rather trivial for the locally compact case, it is not so when local compactness is removed as an assumption. The question remains widely open even for the simplest case of a set of only three points as pointed out by Gromov [1]}. Moreover in the locally compact case it is not clear whether the convex hull itself of a compact set, without taking its closure, is compact.
